Theorem breq12 3987

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 3985	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 3986	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 458	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   class class class wbr 3982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983

This theorem is referenced by:  breq12i  3991  breq12d  3995  breqan12d  3998  posng  4676  isopolem  5790  poxp  6200  rbropapd  6210  ecopover  6599  ecopoverg  6602  ltdcnq  7338  recexpr  7579  ltresr  7780  reapval  8474  ltxr  9711  xrltnr  9715  xrltnsym  9729  xrlttr  9731  xrltso  9732  xrlttri3  9733  xposdif  9818  exmidsbthrlem  13901